What you say may more or less apply to a special situation in the UK, especially England, with its relics of a strongly divided social class society.

I was only really concerned with whether or not public schools in UK were and are named 'public' for sensible reason that we can understand-they are, that is the case..By the way this is not just an English term [public school] since public schools in Scotland were always known as public schools; although, their local government; shortly to be their Independent national Scottish government, may have, or maybe will change that. Can you legislate for what people call things though?